Approximation of symmetrizations by Markov processes

TL;DR

This paper demonstrates that under certain conditions, Markov processes formed by successive symmetrizations converge to a symmetrization, with applications to various geometric rearrangements.

Contribution

It introduces a framework showing convergence of symmetrization Markov processes, including new results for spherical rearrangements and a quantitative asymmetry measure.

Findings

Markov process convergence to symmetrization

Applicability to Steiner, polarization, and cap symmetrizations

Quantitative asymmetry measure as a key tool

Abstract

Under continuity and recurrence assumptions, we prove that the iteration of successive partial symmetrizations that form a time-homogeneous Markov process, converges to a symmetrization. We cover several settings, including the approximation of the spherical nonincreasing rearrangement by Steiner symmetrizations, polarizations and cap symmetrizations. A key tool in our analysis is a quantitative measure of the asymmetry.